Zauner's conjecture: do SIC-POVMs exist in every dimension?

Numerically found in every dimension tried, exactly constructed in many, and proved in none in general.

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Problem statement

A symmetric informationally complete positive operator-valued measure in dimension $d$ is a set of $d^2$ unit vectors $\{|\psi_j\rangle\}$ in $\mathbb{C}^d$ with

|\langle \psi_j | \psi_k \rangle|^2 = \frac{1}{d+1} \quad \text{for all } j \ne k.

Conjecture: such a set exists for every $d \ge 2$ , and can be taken as an orbit of the Weyl–Heisenberg group.

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Background

Posed by Gerhard Zauner in his 1999 thesis and independently rediscovered by Renes, Blume-Kohout, Scott and Caves. The problem has an unexpected connection to Hilbert's twelfth problem: the exact solutions generate abelian extensions of real quadratic fields, and work by Appleby, Flammia, Kopp and others has turned SIC construction into a question in algebraic number theory.

Why it matters

SIC-POVMs are optimal measurements for quantum state tomography and minimal informationally complete measurements. Beyond quantum information, the number-theoretic reformulation makes this a rare problem where a physics question feeds directly into class field theory.

Known partial results

Exact solutions are known in dimensions $2$ – $28$ and a scattered collection of larger dimensions, some found with high-precision numerics and then recognised algebraically.
Numerical solutions have been found in every dimension attempted, into the hundreds.
Kopp and collaborators have given conjectural exact constructions from Stark units, verified in many cases.
No proof exists for infinitely many dimensions.

References

[1]Randomstrasse101 — Open Problems of 2025 — Mutually Unbiased Bases, ETFs, and Zauner’s Conjecture — Problem 24 link

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